I have been dipping into Badiou's 1988 magnum opus
"Being and Event", in the 2005 translation by
Oliver Feltham.
The parts on logic and mathematics that I examined
seemed quite competently written, and Badiou has
done his homework -- in particular, he seems
to have studied Kunen's textbook on set theory
fairly closely, as well as other sources.
Nevertheless, the book is very definitely in the
tradition of speculative philosophy that starts
with Plato's "Parmenides", and continues with Hegel,
Heidegger, Sartre, Lacan, Derrida and others.
This tradition, notable for its impenetrable prose,
is the currently dominant mode in French philosophy.
Like most philosophers in this tradition, Badiou is
enthralled by the One and the Many and also by the
Nothing. Several chapters are devoted to this last
topic. In Meditation 5, where the axioms of set
theory are discussed, no less than 3 pages are devoted
to the discussion of the null set axiom. Here is
how Badiou explains it:
In its metaontological formulation the axiom says:
the unpresentable is presented, as a subtractive
term of the presentation of a presentation.
Or: being lets itself be named, within the ontological
situation, as that from which existence does not exist
(p. 67-68).
Here is his explanation of why the null set is unique:
What ensures the uniqueness of the void-set is that in
wishing to think of it as a species or a common name,
in supposing that there are `several' voids, I expose
myself within the framework of the ontological theory
of the multiple, to the risk of overthrowing the regime
of the same and the other, and so to *having to found
difference on something other than belonging*. Yet
any such procedure is equivalent to restoring the being
of the one (p. 68).
These kind of neo-Hegelian "explanations" are entirely typical
of the treatise. I haven't chosen particularly obscure passages --
there are worse.
A surprising feature of the book is the central place given
to forcing and the notion of genericity. The importance of
these concepts for Badiou is (if I understand him correctly) that
they form a model for the development of new notions over time,
an idea that is central to Beth's and Kripke's semantics for
intuitionism, though I am not sure if Badiou is aware of this. This
emerges fairly clearly in Meditation 35 (p. 398), where Badiou's earlier
obscure remarks about the generic are clarified.
Like most of the works in this tradition, Badiou's book has
a dreamlike quality, like the ideas that float in your head
as you are falling asleep. Long passages seem to hover on
the verge of comprehensibility.
For any readers of FOM who would like to see how modern
set theory appears to a talented French philosopher steeped
in the works of Plato, Hegel, Heidegger, Marx, Lenin,
Mallarm'e, Mao, Lacan etc., I can recommend the book strongly.
Personally, I prefer my set theory straight.
Alasdair Urquhart